Center for Computational and Stochastic Mathematics

Publicações

A meshfree method with domain decomposition for Helmholtz boundary value problems

Publication . Valtchev, Svilen

In the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.

2020-11Comunicação em conferência

Acesso aberto

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Solving boundary value problems on manifolds with a plane waves method

Publication . Alves, Carlos J. S.; Antunes, Pedro R. S.; Martins, Nuno F. M.; Valtchev, Svilen S.

In this paper we consider a plane waves method as a numerical technique for solving boundary value problems for linear partial differential equations on manifolds. In particular, the method is applied to the Helmholtz–Beltrami equations. We prove density results that justify the completeness of the plane waves space and justify the approximation of domain and boundary data. A-posteriori error estimates and numerical experiments show that this simple technique may be used to accurately solve boundary value problems on manifolds.

2020-09Artigo científico

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Meshfree Approximate Solution of the Cauchy-Navier Equations of Elastodynamics

Publication . Valtchev, Svilen S.; Martins, Nuno F. M.

In this study we propose a meshfree scheme for the numerical solution of boundary value problems (BVP) for the nonhomogeneous Cauchy-Navier equations of elastodynamics. The method uses the classical approach where first a particular solution for the partial differential equation (PDE) is calculated and then the corresponding homogeneous BVP is solved for the homogeneous part of the total solution. In particular, we approximate each component of the source term of the nonho-mogeneous PDE by superposition of plane wave functions with different frequencies and directions of propagation. Using these expansions, a particular solution for the PDE is derived in the form of a linear combination of elastic P-And S-waves, at no extra computational cost. In the second step of the scheme, we solve the corresponding homogeneous BVP using the classical method of fundamental solutions (MFS), with shape functions given by the Kupradze tensor. The accuracy and the convergence of the proposed technique is illustrated for a Dirichlet BVP, posed in a 2D multiply connected domain, bounded by polygonal and parametric curves.

2021-11Comunicação em conferência

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Palavras-chave

Algebra and Computing,Applied and Numerical Analysis,Mathematical Modelling in Biomedicine,Statistics and Stochastic Processes,

Entidade financiadora

Fundação para a Ciência e a Tecnologia, I.P.
Fundação para a Ciência e a Tecnologia, I.P.

Programa de financiamento

6817 - DCRRNI ID
Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017/2018) - Financiamento Base

Número da atribuição

UIDB/04621/2020